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Tracing the motion of electrons has enormous relevance to understandingubiquitous phenomena in ultrafast science, such as the dynamical evolution ofthe electron density during complex chemical and biological processes.Scattering of ultrashort x-ray pulses from an electronic wavepacket wouldappear to be the most obvious approach to image the electronic motion inreal-time and real-space with the notion that such scattering patterns, in thefar-field regime, encode the instantaneous electron density of the wavepacket.However, recent results by Dixit {\em et al.} [Proc. Natl. Acad. Sci. U.S.A.,{\bf 109}, 11636 (2012)] have put this notion into question and shown that thescattering in the far-field regime probes spatio-temporal density-densitycorrelations. Here, we propose a possible way to image the instantaneouselectron density of the wavepacket via ultrafast x-ray {\em phase contrastimaging}. Moreover, we show that inelastic scattering processes, which plagueultrafast scattering in the far-field regime, do not contribute in ultrafastx-ray phase contrast imaging as a consequence of an interference effect. Weillustrate our general findings by means of a wavepacket that lies in the timeand energy range of the dynamics of valence electrons in complex molecular andbiological systems. This present work offers a potential to image not onlyinstantaneous snapshots of non-stationary electron dynamics, but also theLaplacian of these snapshots which provide information about the complexbonding and topology of the charge distributions in the systems.

Proposed imaging of the ultrafast electronic motion in samples using x-ray phase-contrast